Advertisement

Archive for Rational Mechanics and Analysis

, Volume 207, Issue 1, pp 303–316 | Cite as

Serrin-Type Blowup Criterion for Full Compressible Navier–Stokes System

  • Xiangdi Huang
  • Jing LiEmail author
  • Yong Wang
Article

Abstract

The authors establish a Serrin-type blowup criterion for the Cauchy problem of the three-dimensional full compressible Navier–Stokes system, which states that a strong or smooth solution exists globally, provided that the velocity satisfies Serrin’s condition and that the temporal integral of the maximum norm of the divergence of the velocity is bounded. In particular, this criterion extends the well-known Serrin’s blowup criterion for the three-dimensional incompressible Navier–Stokes equations to the three-dimensional full compressible system and is just the same as that of the barotropic case.

Keywords

Weak Solution Cauchy Problem Stokes Equation Strong Solution Smooth Solution 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Beale J.T., Kato T., Majda A.: Remarks on the breakdown of smooth solutions for the 3-D Euler equations. Commun. Math. Phys. 94, 61–66 (1984)MathSciNetADSzbMATHCrossRefGoogle Scholar
  2. 2.
    Cho Y., Kim H.: Existence results for viscous polytropic fluids with vacuum. J. Differ. Equ. 228, 377–411 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Fan J., Jiang S., Ou Y.: A blow-up criterion for compressible viscous heat-conductive flows. Annales de l’Institut Henri Poincare (C) Analyse non lineaire 27, 337–350 (2010)MathSciNetADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Feireisl E.: Dynamics of Viscous Compressible Fluids. Oxford Science Publication, Oxford (2004)zbMATHGoogle Scholar
  5. 5.
    Feireisl E., Novotny A., Petzeltová H.: On the existence of globally defined weak solutions to the Navier–Stokes equations. J. Math. Fluid Mech. 3, 358–392 (2001)MathSciNetADSzbMATHCrossRefGoogle Scholar
  6. 6.
    Hoff D.: Global solutions of the Navier–Stokes equations for multidimensional compressible flow with discontinuous initial data. J. Differ. Equ. 120(1), 215–254 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Hoff D.: Discontinuous solutions of the Navier–Stokes equations for multidimensional flows of heat-conducting fluids. Arch. Rational Mech. Anal. 139, 303–354 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Huang, X.D.: Some results on blowup of solutions to the compressible Navier–Stokes equations. PhD thesis, The Chinese University of Hong Kong (2009)Google Scholar
  9. 9.
    Huang X.D., Li J.: On breakdown of solutions to the full compressible Navier–Stokes equations. Methods Appl. Anal. 16, 479–490 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Huang, X.D., Li, J.: Global classical and weak solutions to the three-dimensional full compressible Navier–Stokes system with vacuum and large oscillations. http://arxiv.org/abs/1107.4655
  11. 11.
    Huang X.D., Li J., Xin Z.P.: Serrin type criterion for the three-dimensional viscous compressible flows. SIAM J. Math. Anal. 43, 1872–1886 (2011)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Huang X.D., Li J., Xin Z.P.: Blowup criterion for viscous barotropic flows with vacuum states. Commun. Math. Phys. 301, 23–35 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  13. 13.
    Huang X.D., Li J., Xin Z.P.: Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier–Stokes equations. Commun. Pure Appl. Math. 65, 549–585 (2012)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Huang X.D., Xin Z.P.: A blow-up criterion for classical solutions to the compressible Navier–Stokes equations. Sci. China 53, 671–686 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Kazhikhov A.V.: Cauchy problem for viscous gas equations. Sib. Math. J. 23, 44–49 (1982)MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Kazhikhov A.V., Shelukhin V.V.: Unique global solution with respect to time of initial-boundary value problems for one-dimensional equations of a viscous gas. J. Appl. Math. Mech. 41, 273–282 (1977)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Lions P.L.: Mathematical Topics in Fluid Mechanics, vol. 2. Compressible Models. Oxford University Press, New York (1998)zbMATHGoogle Scholar
  18. 18.
    Matsumura A., Nishida T.: The initial value problem for the equations of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Nash J.: Le problème de Cauchy pour les équations différentielles d’un fluide général. Bull. Soc. Math. France 90, 487–497 (1962)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Nirenberg L.: On elliptic partial differential equations. Ann. Scuola Norm. Sup. Pisa 13(3), 115–162 (1959)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Rozanova O.: Blow up of smooth solutions to the compressible Navier–Stokes equations with the data highly decreasing at infinity. J. Differ. Equ. 245, 1762–1774 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Serrin J.: On the uniqueness of compressible fluid motion. Arch. Rational Mech. Anal. 3, 271–288 (1959)MathSciNetADSzbMATHCrossRefGoogle Scholar
  23. 23.
    Serrin J.: On the interior regularity of weak solutions of the Navier–Stokes equations. Arch. Rational Mech. Anal. 9, 187–195 (1962)MathSciNetADSzbMATHCrossRefGoogle Scholar
  24. 24.
    Sun Y.Z., Wang C., Zhang Z.F.: A Beale-Kato-Majda blow-up criterion for the 3-D compressible Navier–Stokes equations. J. Math. Pures Appl. 95, 36–47 (2011)MathSciNetzbMATHGoogle Scholar
  25. 25.
    Sun Y.Z., Wang C., Zhang Z.F.: A Beale-Kato-Majda criterion for three dimensional compressible viscous heat-conductive flows. Arch. Rational Mech. Anal. 201, 727–742 (2011)MathSciNetADSzbMATHCrossRefGoogle Scholar
  26. 26.
    Xin Z.P.: Blowup of smooth solutions to the compressible Navier–Stokes equation with compact density. Commun. Pure Appl. Math. 51, 229–240 (1998)zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.NCMIS, AMSSChinese Academy of SciencesBeijingPeople’s Republic of China
  2. 2.Department of Pure and Applied MathematicsGraduate School of Information Science and Technology, Osaka UniversityToyonakaJapan
  3. 3.Institute of Applied Mathematics & NCMIS, AMSS, and Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingPeople’s Republic of China
  4. 4.Academy of Mathematics and Systems ScienceChinese Academy of SciencesBeijingPeople’s Republic of China

Personalised recommendations